Chapter 7.  Control Charts for Attributes
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Chapter 7. Control Charts for Attributes. Control Chart for Fraction Nonconforming. Fraction nonconforming is based on the binomial distribution. n : size of population p : probability of nonconformance D : number of products not conforming Successive products are independent.

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Chapter 7. Control Charts for Attributes

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Chapter 7 control charts for attributes

Chapter 7. Control Charts for Attributes


Chapter 7 control charts for attributes

Control Chart for Fraction Nonconforming

Fraction nonconforming is based on the binomial distribution.

n: size of population

p: probability of nonconformance

D: number of products not conforming

Successive products are independent.

Mean of D = np

Variance of D = np(1-p)


Chapter 7 control charts for attributes

Sample fraction nonconformance


Chapter 7 control charts for attributes

w: statistics for quality

Mean of w: μw

Variance of w: σw2

L: distance of control limit from center line (in standard deviation units)

If p is the true fraction nonconformance:


Chapter 7 control charts for attributes

If p is not know, we estimate it from samples.

m: samples, each with n units (or observations)

Di: number of nonconforming units in sample i

Average of all observations:


Chapter 7 control charts for attributes

Example 6-1. 6-oz cardboard cans of orange juice


Chapter 7 control charts for attributes

If samples 15 and 23 are eliminated:


Chapter 7 control charts for attributes

Additional samples collected after adjustment of control chart:


Chapter 7 control charts for attributes

Control chart variables using only the recent 24 samples:

Set equal to zero for negative value


Design of fraction nonconforming chart

Design of Fraction Nonconforming Chart

Three parameters to be specified:

  • sample size

  • frequency of sampling

  • width of control limits

    Common to base chart on 100% inspection of all process output over time.

    Rational subgroups may also play role in determining sampling frequency.


Chapter 7 control charts for attributes

np Control Chart


Chapter 7 control charts for attributes

Variable Sample Size

Variable-Width Control Limits


Chapter 7 control charts for attributes

Variable Sample Size

Control Limits Based on an Average Sample Size

Use average sample size. For previous example:


Chapter 7 control charts for attributes

Variable Sample Size

Standard Control Chart

- Points are plotted in standard deviation units.

UCL = 3

Center line = 0

LCL = -3


Chapter 7 control charts for attributes

Skip Section 6.2.3 pages 284 - 285


Chapter 7 control charts for attributes

Operating Characteristic Function and Average Run Length Calculations

Probability of type II error


Chapter 7 control charts for attributes

Since D is an integer,


Chapter 7 control charts for attributes

Average run length

If the process is in control:

If the process is out of control


Chapter 7 control charts for attributes

Control Charts for Nonconformities (or Defects)

Procedures with Constant Sample Size

x: number of nonconformities

c > 0: parameter of Poisson distribution

Set to zero if negative


Chapter 7 control charts for attributes

If no standard is given, estimate c then use the following parameters:

Set to zero if negative


Chapter 7 control charts for attributes

There are 516 defects in total of 26 samples. Thus.


Chapter 7 control charts for attributes

There are 516 defects in total of 26 samples. Thus.

Sample 6 was due to inspection error.

Sample 20 was due to a problem in wave soldering machine.

Eliminate these two samples, and recalculate the control parameters.

New control limits:


Chapter 7 control charts for attributes

Additional samples collected.


Chapter 7 control charts for attributes

Further Analysis of Nonconformities


Chapter 7 control charts for attributes

Choice of Sample Size: μ Chart

x: total nonconformities in n inspection units

u: average number of nonconformities per inspection unit


Chapter 7 control charts for attributes

Control Charts for Nonconformities

Procedure with Variable Sample Size


Chapter 7 control charts for attributes

Control Charts for Nonconformities

Demerit Systems: not all defects are of equal importance


Chapter 7 control charts for attributes

ciA: number of Class A defects in ith inspection units

Similarly for ciB, ciC, and ciD for Classes B, C, and D.

di: number of demerits in inspection unit i

Constants 100, 50, 10, and 1 are demerit weights.


Chapter 7 control charts for attributes

µi: linear combination of independent Poisson variables


Chapter 7 control charts for attributes

Control Charts for Nonconformities

Operating Characteristic Function

x: Poisson random variable

c: true mean value

β: type II error probability


Chapter 7 control charts for attributes

For example 6-3

Number of nonconformities is integer.


Chapter 7 control charts for attributes

Control Charts for Nonconformities

Dealing with Low Defect Levels

  • If defect level is low, <1000 per million, c and u charts become ineffective.

  • The time-between-events control chart is more effective.

  • If the defects occur according to a Poisson distribution, the probability distribution of the time between events is the exponential distribution.

  • Constructing a time-between-events control chart is essentially equivalent to control charting an exponentially distributed variable.

  • To use normal approximation, translate exponential distribution to Weibull distribution and then approximate with normal variable


Chapter 7 control charts for attributes

Guidelines for Implementing Control Charts

Applicable for both variable and attribute control


Chapter 7 control charts for attributes

Determining Which Characteristics and

Where to Put Control Charts


Chapter 7 control charts for attributes

Choosing Proper Type of Control Chart


Chapter 7 control charts for attributes

Actions Taken to Improve Process


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